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Primes and the Fundamental Theorem of Arithmetic

Primes are the atoms of multiplication. Every integer above 1 breaks into primes in exactly one way — and that uniqueness is a theorem worth its grand name.

Prime, composite, and the odd case of 1

A [[alg-prime-number|prime number]] is an integer greater than 1 whose only positive factors are 1 and itself: 2, 3, 5, 7, 11, 13, … . Anything greater than 1 that is not prime is [[composite-number|composite]] — it factors further, like 6 = 2·3 or 9 = 3·3. The number 1 is deliberately neither prime nor composite; calling it prime would wreck the uniqueness we are about to celebrate.

Breaking a number into its primes

[[prime-factorization|Prime factorization]] writes a composite number as a product of primes. The reliable method is a factor tree: peel off the smallest prime you can, again and again, until only primes remain.

Factor 360:
  360 ÷ 2 = 180
  180 ÷ 2 =  90
   90 ÷ 2 =  45
   45 ÷ 3 =  15
   15 ÷ 3 =   5
    5 is prime — stop.

360 = 2 · 2 · 2 · 3 · 3 · 5
    = 2^3 · 3^2 · 5

Check: 8 · 9 · 5 = 72 · 5 = 360. ✓
Repeated division by the smallest available prime gives 360 = 2^3 · 3^2 · 5.

Why the factorization is unique

The [[fundamental-theorem-of-arithmetic|fundamental theorem of arithmetic]] says every integer greater than 1 is a product of primes in exactly one way, apart from the order of the factors. You will never find one route to 360 that ends in 2^3·3^2·5 and another that ends in 7·something. This is why primes deserve to be called the building blocks of the integers.

Uniqueness is what makes number theory tick. It guarantees that two numbers with no common prime are truly “coprime,” and it lets us read off divisibility, GCFs, and much more directly from the prime list. We will lean on it again and again.